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Reliability assessment of two‐level balanced systems with common bus performance sharing subjected to epistemic uncertainty

August 2023
Quality and Reliability Engineering 40(5)

August 2023
40(5)

DOI:10.1002/qre.3429

Authors:

Tianzi Tian

Beihang University (BUAA)

Jun Yang

Beihang University (BUAA)

Ning Wang

Beihang University (BUAA)

Two‐level balanced systems with common bus performance sharing (TLBS‐CBPS), represented by power battery systems, play an important role in various industries. Where the two‐level balance refers to when the system is unbalanced, redistributing performance between modules using a common bus for the intermodule balance, followed by the in‐module component performance redistribution to achieve global balance, that is, the performance differences among all components are within the balance degree threshold. However, the component performance has epistemic uncertainty due to measurement limitations, and the distance‐dependent transmission loss incurs during the performance redistribution. These problems make the reliability evaluation of TLBS‐CBPS challenging. To address these issues, a reliability evaluation method for TLBS‐CBPS is proposed by considering two‐level balance, epistemic uncertainty, and transmission loss. First, the system model is constructed from working modes, and the two‐level rebalancing process is described as three nonlinear programming problems. Then, based on the Dempster–Shafer evidence theory, the belief universal generating function (BUGF) method is proposed to evaluate the uncertainty of the system reliability. To improve the computation efficiency, a simplified BUGF algorithm is given through merging, judging, labeling, and expanding realizations. Finally, two examples of lithium‐ion battery packs demonstrate the effectiveness of the proposed method.

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Received:  March  Revised:  August  Accepted:  August 

DOI: ./qre.

SPECIAL ISSUE ARTICLE

Reliability assessment of two-level balanced systems with

common bus performance sharing subjected to epistemic

uncertainty

Tianzi Tian Jun Yang Ning Wang

School of Reliability and Systems

Engineering, Beihang University, Beijing,

China

Correspondence

Jun Yang, School of Reliability and

Systems Engineering, Beihang University,

Beijing, China.

Email: tomyj@buaa.edu.cn

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 

Abstract

Two-level balanced systems with common bus performance sharing (TLBS-

CBPS), represented by power battery systems, play an important role in various

industries. Where the two-level balance refers to when the system is unbal-

anced, redistributing performance between modules using a common bus for the

intermodule balance, followed by the in-module component performance redis-

tribution to achieve global balance, that is, the performance differences among

all components are within the balance degree threshold. However, the compo-

nent performance has epistemic uncertainty due to measurement limitations,

and the distance-dependent transmission loss incurs during the performance

redistribution. These problems make the reliability evaluation of TLBS-CBPS

challenging. To address these issues, a reliability evaluation method for TLBS-

CBPS is proposed by considering two-level balance, epistemic uncertainty, and

transmission loss. First, the system model is constructed from working modes,

and the two-level rebalancing process is described as three nonlinear pro-

gramming problems. Then, based on the Dempster–Shafer evidence theory, the

belief universal generating function (BUGF) method is proposed to evaluate

the uncertainty of the system reliability. To improve the computation effi-

ciency, a simplified BUGF algorithm is given through merging, judging, labeling,

and expanding realizations. Finally, two examples of lithium-ion battery packs

demonstrate the effectiveness of the proposed method.

KEYWORDS

belief universal generating function, epistemic uncertainty, reliability assessment, transmis-

sion loss, two-level balanced systems

1 INTRODUCTION

Balanced systems originated from space propulsion and military equipment such as planetary lander systems and multi-

rotor drone systems. Nowadays, an increasing number of balanced systems is studied for its reliability, such as unmanned

aerial vehicle systems,– port systems,communication systems,and so on.– The power battery system, as a two-level

balanced system with common bus performance sharing (TLBS-CBPS), often serves as a critical part of many devices.

Once an imbalance or failure occurs, it may lead to mission failure or significant economic losses. However, most existing

Qual Reliab Engng Int. ;–. ©  John Wiley & Sons Ltd. 1wileyonlinelibrary.com/journal/qre

2TIAN  .

works in reliability modeling and analysis only focus on the one-level balanced topology. There has been little attention

paid to the two-level balanced topology. Meanwhile, the battery performance has epistemic uncertainty and transmis-

sion losses occur during the performance transmission. Therefore, the reliability assessment of TLBS-CBPS considering

two-level balance, epistemic uncertainty, and transmission loss is studied in this work.

In , Hua and Elsayed, introduced the concept of balance in the reliability area for the first time, where the system

reliability not only depends on the reliability of individual units but also on whether their configuration is balanced.

Further, Cui et al. investigated the reliability of m-sector k-out-of-n: F balanced systems, where the balance means that

each sector always has an equal number of working components. The concept of balance, in which the number of working

components is the same or has a certain range of differences, has been explored by Zhao et al. and Wang et al. In

addition, some scholars have studied balanced systems that system failure occurs if the difference in component states

exceeds a threshold.,,

To improve the reliability of balanced systems, when a balanced system fails due to losing balance, rebalancing opera-

tions are usually employed to restore the system balance. Cui and Fangused the engine start-up rules as a rebalancing

mechanism and enriched the reliability analysis of m-sector balanced engine systems by considering the start-up proba-

bility. Wang and Miao considered maintenance operations as rebalancing operations, where preventive maintenance is

performed once the state difference between two symmetric components exceeds the threshold to avoid system failure.

In addition, Wang et al. incorporated the triggering mechanism of protective devices into the rebalancing mechanism,

where protective devices can mitigate the component degradation.

Currently, with the development of new energy technologies, battery systems have integrated performance sharing

systems and balanced systems, forming a type of balanced system, namely, the performance-based balanced system

with common bus performance sharing (PBSs-CBPS). The concept of balance in this system has also developed from ini-

tially considering only the performance balance between supply and demand, to considering the performance balance

between components. Tian et al. proposed a reliability model of PBSs-CBPS that considers the performance transmis-

sion loss and balance degree threshold based on battery systems, but they only take into account the performance sharing

between components.

In this study, we discuss a novel balanced system, a lithium-ion battery system with a two-level balanced architec-

ture, that is, TLBS-CBPS. Battery cells in a pack may have voltage differences due to different usage states and lifetimes,

which can cause over-charging, over-discharging, and even failure of the pack. Balanced technology aims to maintain

cell balance by controlling charge transfer, and improving the reliability of the battery pack. In applications with many

series-connected battery modules, the balancing speed of a single-layer series topology is limited by the capacity of a single

balancer on the balancing path. To improve the speed and efficiency, a two-level balance architecture was proposed and

analyzed in Refs., In this study, we investigated TLBS-CBPS, which consists of nmodules, with each module contain-

ing mcomponents of the same type and specification. The two-level rebalancing mechanism is the performance transfer

between cells in different modules via the common bus. The “two-level” means that the first level balance achieves per-

formance balance between modules, while the second level achieves performance balance among components within

each module. Transmission losses and capacity constraints exist in performance transfer, and the transmission loss rate

between modules is affected by distance.

In practical engineering, the performance level of individual cells is estimated based on parameters such as voltage,

current, and temperature, resulting in uncertain outcomes.,, Such uncertainty caused by incomplete knowledge

and incomplete information is referred to as epistemic uncertainty., Currently, many methods have been proposed

to address reliability assessment problems of complex systems under epistemic uncertainty, such as Dempster–Shafer evi-

dence theory (DSET), fuzzy set theory, and interval theory. Meanwhile, the reliability assessment methods have also

matured and can be broadly classified into two categories:

() reliability assessment methods based on stochastic processes, such as Markov models, aggregated stochastic

processes, and finite Markov embedding chains,;

() reliability assessment methods based on probability statistics, such as the LaPlace method, universal generating

function (UGF) method, Bayes method, and so on,, where the UGF method has great advantages in deriving the

reliability of performance-sharing systems.

However, there is currently no research on the reliability assessment method of balanced systems with epistemic

uncertainty, two-level architecture, and distance-dependent transmission loss. This study proposes a reliability model

for TLBS-CBPS, which is a dynamic-balanced system that distributes performance between modules and components to

maintain balance. First, considering the epistemic uncertainty of the components’ performance level, this study quantifies

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TIAN  . 3

FIGURE 1 The motivation and framework of our work.

the epistemic uncertainty using the DSET, and a modified Markov model combined with mass functions is proposed to

obtain precise confidence levels of component states. Second, this study fills the research gap in the reliability model of

balanced systems with a two-level balanced topology. Unlike previous studies that assumed fixed transmission loss, the

transmission loss between modules is distance-dependent. Then, to model the two-level rebalancing process, three nonlin-

ear programming models are built. Finally, the belief universal generating function (BUGF) method is applied to evaluate

the epistemic uncertainty of the system reliability, with a simplified algorithm to address the combinatorial explosion

problem. In addition, the results of the sensitivity analysis on the influencing factors provide guidance for improving the

TLBS-CBPS design and configuration to enhance system reliability. To better comprehend the innovation and significance

of this study, Figure presents the motivation and framework of our work.

In summary, the main contributions of this study are as follows:

. To characterize the two-level rebalancing process of TLBS-CBPS, the performance balancing among modules, the

performance transfer between modules and components, and the performance balancing among components are,

respectively, formulated as three nonlinear programming problems to obtain the performance of components after

rebalancing.

. Considering the epistemic uncertainty and complex transmission of component performance, a novel BUGF method

is proposed to evaluate the reliability of TLBS-CBPS, where system operation conditions and nonlinear programming

solutions are converted into indicator functions for calculation.

. To alleviate the combinatorial explosion problem of BUGF, a simplified BUGF algorithm is given by merging, judging,

labeling, and expanding realizations to improve the computational efficiency.

4TIAN  .

FIGURE 2 The diagram of the two-level balanced architecture of battery systems.

The organization of this study is as follows. Section provides a detailed description of the two-level rebalancing process

and the construction of the reliability model of TLBS-CBPS. Section proposed a reliability assessment method of TLBS-

CBPS based on the BUGF and presents a simplified algorithm to alleviate the combinatorial explosion problem. Section 

provides two examples to verify the applicability of the proposed model. Section discusses the conclusions and future

research directions.

2TWO-LEVEL BALANCED SYSTEM RELIABILITY MODEL

This section describes the TLBS-CBPS and focuses on the two-level rebalancing process. Furthermore, a reliability model

for TLBS-CBPS is presented, where the rebalancing process is transformed into nonlinear programming problems.

2.1 Two-level balanced system description

Taking the battery pack as an example, the architecture of the two-level balanced system is shown in Figure , including

nine individual cells. The system comprises nine individual cells, organized into three cells connected in series to form a

battery module, and three battery modules connected in series to form a battery pack, resulting in a two-level architecture.

Each module has a module management unit (MMU), which can monitor parameters and communicate with the central

management unit (CMU) via the controller area network (CAN) bus. Based on voltage and bus current data collected by

the CMU, the MMU can estimate the performance of each cell, although some estimation error is inevitable. Then, the

CMU can collect all the performance values of the battery pack via the CAN bus.

When the system is out of balance, the CMU sends driving signals to initiate the rebalancing operation. The mean

difference-average (MDA) method is typically used in two-level balanced systems for rebalancing. The rebalancing

operation is divided into two stages.

In the first stage, the first-level balance is executed by a separate modular balancer (IME) and CMU, transferring per-

formance from modules with higher average performance to those with lower average performance. The criterion for

stopping the first-level balance is that the average performance difference between modules (𝛿1) is less than the balance

threshold (𝜀1) between modules.

In the second stage, the second-level balance is executed to balance components until the difference between the com-

ponents’ performance and the average performance (𝛿2) converge to the balance degree threshold between components

(𝜀2). Balancing of these components can be efficiently performed by an individual cell equalizer (ICE) and MMU.

TIAN  . 5

FIGURE 3 The simplified diagram of the two-level balanced architecture of battery systems.

To understand the rebalancing process more clearly, we give the simplified diagram of the two-level balanced

architecture of the battery system in Figure .

Adjacent modules are connected through IMEs, which use performance storage devices like transformers and capacitors

(the red nodes in Figure ) to transfer performance. However, nonadjacent modules must use the middle storage device

as a transmission medium, and the transmission performance loss will increase as the distance between modules grows.

This is because performance must be transferred through more mediums, and charging and discharging energy storage

devices will cause performance losses. On the other hand, every two cells are both connected by ICE, and the performance

transmission of any two cells can be realized by ICE and MMU without the middle medium. Meanwhile, the performance

can be transferred either from component to module or module to component by ICE. The blue arrow in Figure shows

how all modules can share performance through a common bus. Similarly, all cells can be regarded as connected by a

common bus.

Based on engineering practice, this study establishes an operational criterion for the system. The criterion specifies that

the system operates when the performance differences between components are within the designated threshold, and

the total system performance (𝑆) meets the demand performance (𝐷).To facilitate a comprehensive grasp of the system’s

working mode and rebalancing process, the system operational flow chart is graphically presented in Figure .

2.2 System reliability model

First, the random performance of the component 𝑗in module 𝑖over time 𝑡is denoted by 𝑔𝑖,𝑗(𝑡). We define 𝐆𝑖(𝑡) as the

performance set of components in module 𝑖at time 𝑡,𝐆(𝑡) as the performance set of all modules, and 𝑔(𝑡) as the perfor-

mance set of all components in the system. The proposed two-level balanced system includes 𝑛modules, and each module

contains 𝑚components. The number of components in the system is 𝑛1.

𝐆𝑖(𝑡)={𝑔

𝑖,1(𝑡), 𝑔𝑖,2(𝑡), … , 𝑔𝑖,𝑚(𝑡 )}𝑖 = 1, 2, … , 𝑛. ()

𝐆(𝑡) = {𝐺1(𝑡), 𝐺2(𝑡), … , 𝐺𝑛(𝑡)}, ()

where 𝐺𝑖(𝑡) = 𝑠𝑢𝑚(𝐆𝑖(𝑡)).

𝐠(𝑡) = 𝑔1,1(𝑡), 𝑔1,2(𝑡), … , 𝑔1,𝑚 (𝑡), … ., 𝑔𝑛,1(𝑡), 𝑔𝑛,2(𝑡), … , 𝑔𝑛,𝑚(𝑡)()

6TIAN  .

FIGURE 4 The rebalancing operation and working mode of the two-level balanced system.

With the series structure of the battery pack, the performance of the system 𝑆(𝑡) takes the form,

𝑆(𝑡) = min

𝑖=1,2,…𝑛;𝑗=1,2,…,𝑚 𝑔𝑖,𝑗 (𝑡).()

According to the MDA method, the balance degree between modules 𝛿1and balance degree between components in

module 𝑖,𝛿2, can be expressed by

𝛿1(𝑡) = 



𝐆(𝑡) − 𝐺(𝑡)



∞()

𝛿2(𝑡) = 



𝐠(𝑡) − 𝑔(𝑡)



∞()

where AptCommand2016;∞is the infinity norm. 𝐺(𝑡) denotes the average of modules’ performance. 𝐆(𝑡) − 𝐺(𝑡)∞

means the highest absolute value of the difference between 𝐆and 

𝐺. The allowable maximum balance degree threshold

of modules and in-module components are 𝜀1and 𝜀2, respectively.

To improve the efficiency of the rebalancing process, it is essential to minimize the loss of performance that may occur

during the rebalancing operation. Meanwhile, to avoid the frequent operation of rebalancing switches, it is assumed that

the duration of the rebalancing switches’ state should not be lower than a constant. To simplify this requirement, we

define that a transmission capacity limit exists for the common bus, where the intermodule transmission capacity limit is

defined as a constant 𝐶1and the intercomponent transmission capacity limit is defined as a constant 𝐶2.

TIAN  . 7

Subsequently, when 𝛿2(𝑡) > 𝜀2, the system needs to perform the rebalancing operation, and the rebalancing operation

between modules (the first level balance) is modeled using the following nonlinear programming problem.

min

𝑛



𝑘=1

𝑛



𝑙=1

𝜌𝑘𝛽1

𝑘,𝑙𝑥𝑘,𝑙 ()

and subject to the following:

𝛽1

𝑘,𝑙 =𝛼𝑑

𝑘,𝑙, ∀𝑘 = 1, 2, … , 𝑛; 𝑙 = 1, 2, … , 𝑛 ()

𝑥𝑘,𝑘 = 0, ∀𝑘 = 1, 2, … , 𝑛 ()

𝑥𝑘,𝑙 =0,𝑖𝑓𝜌

𝑘= 1, ∀𝑘 = 1, 2, … , 𝑛; 𝑙(𝑙 ≠𝑘)= 1,2,…,𝑛;𝜌

𝑘∈{0, 1}()

𝑥𝑙,𝑘 =0,𝑖𝑓𝜌

𝑘= 0, ∀𝑘 = 1, 2, … , 𝑛; 𝑙(𝑙 ≠𝑘)= 1,2,…,𝑛;𝜌

𝑘∈{0, 1}()

𝑥𝑘,𝑙 ≥0,∀𝑘 = 1,2,…,𝑛;𝑙 = 1,2,…,𝑛 ()

𝑥𝑏

𝑘+

𝑚



𝑙=1

(1 − 𝛽1

𝑘,𝑙)𝑥𝑙,𝑘 =𝑥

𝑎

𝑘,𝑖𝑓𝜌

𝑘= 1, ∀𝑘 = 1, 2, … , 𝑛 ()

𝑥𝑏

𝑘−

𝑛



𝑙=1

𝑥𝑘,𝑙 =𝑥

𝑎

𝑘,𝑖𝑓𝜌

𝑘= 0, ∀𝑘 = 1, 2, … , 𝑛 ()



𝑥𝑎

𝑘

𝑛1

−

𝑚



𝑘=1

𝑥𝑎

𝑘

𝑛1𝑛

≤𝜀1, ∀𝑘 = 1, 2, … , 𝑛 ()

𝑥𝑘,𝑙 ≤𝐶1, ∀𝑘 = 1, 2, … , 𝑛 ()

where Equation () is the objective function, indicating the minimum transmission loss. In Equation (), 𝛼is transmis-

sion loss rate of each pass through the performance storage device, and 𝑑𝑘,𝑙 is the number of performance storage devices

between module kand module l. Then, 𝛽1

𝑘,𝑙 is the transmission loss between module kand module l.𝑥𝑘,𝑙 is defined as

the transmission performance from module kto module l.Equation() prohibits self-transfer. Equation ()indicates

that when module kreceives the performance from other modules, it cannot transfer it to other modules, while Equa-

tion () indicates that once module ktransfers performance to other modules, it cannot receive any performance from

them. Here, the binary variable 𝜌𝑘indicates whether the module kreceives ( 𝜌𝑘=1) or outputs ( 𝜌𝑘=0) the perfor-

mance. Equation () introduces a constraint on the performance of the transmission, requiring it to be non-negative. In

Equation (), 𝑥𝑏

𝑖is module 𝑖’s performance before rebalancing, and 𝑥𝑎

𝑖is its performance after rebalancing. Equations ()

and (), respectively, describe module output and input performance update rules considering transmission loss. Equa-

tion () represents when a module outputs performance, its subsequent performance is its original performance minus

the output performance. Equation () describes when a module inputs performance, its subsequent performance is its

original performance plus the input performance after transmission loss. Equation () imposes a constraint that 𝛿1is less

𝜀1, and Equation () states that the performance outputted by a given module must not surpass 𝐶1.

After the first-level balance, each module’s MMU can obtain the performance of its module. Then, to acquire the per-

formance of components after the first-level balance, a nonlinear programming problem is adopted to characterize the

transmission performance process.

min max(𝐲𝑎

𝑖)−𝑚𝑒𝑎𝑛(𝐲

𝑎

𝑖)()

and subject to the following:

𝐲𝑎

𝑖=𝑦𝑎

𝑖,1,…,𝑦

𝑎

𝑖,𝑚, ∀𝑖 = 1, 2, … , 𝑛 ()

8TIAN  .

𝑦𝑏

𝑖,𝑗 +Δ

𝑖,𝑗 =𝑦

𝑎

𝑖,𝑗, ∀𝑖 = 1, 2, … , 𝑛; 𝑗 = 1, 2, … ., 𝑚 ()

𝑚



𝑗=1

Δ𝑖,𝑗 =𝑥

𝑎

𝑖−𝑥

𝑏

𝑖, ∀𝑖 = 1, 2, … , 𝑛 ()

Δ𝑖,𝑗 ⋅(𝑥

𝑎

𝑖−𝑥

𝑏

𝑖)≥0,∀𝑖 = 1,2,…,𝑛;𝑗 = 1,2,….,𝑚 ()

where the performance of component 𝑗in module 𝑖before and after rebalancing between modules is represented by 𝑦𝑏

𝑖,𝑗

and 𝑦𝑎

𝑖,𝑗, respectively. Δ𝑖,𝑗 is the performance of transmission between module 𝑖and in-module component 𝑗, where Δ𝑖,𝑗 is

non-negative when the transfer performance direction is module-to-component and negative. 𝐲𝑎

𝑖is the set of component

performances in module 𝑖after the first-level balance, that is, Equation (). The objective function aims to minimize

the performance difference between in-module components. Equation () gives the performance of components after

rebalancing, obtained by adding the transmission performance to its original performance. Furthermore, Equation ()

means the sum of the output or input performance of the components within module 𝑖is identical to the input or output

performance of module 𝑖during the first-level balance. Equation () represents if a module receives performance during

the first-level balance, performance should be transferred from the module towards components, while a module outputs

performance, performance should be transferred from components towards the module.

Next, if the balance degree between components is still higher than 𝜀2after the first-level balance, the second-level

balance needs to be performed. Then, the rebalancing operation between in-module components is modeled using the

following nonlinear programming problem:

min

𝑚



𝑖=1

𝑛1



𝑗=1

2𝛽2𝑧𝑖,𝑗()

and subject to the following:

𝑚



𝑗=1

𝑧𝑖,𝑗 = 0, ∀𝑖 = 1, 2, … , 𝑛 ()

𝑧𝑏

𝑖,𝑗 =𝑧

𝑎

𝑖,𝑗 +𝑧

𝑖,𝑗,𝑖𝑓𝑧

𝑖,𝑗 < 0, ∀𝑖 = 1, 2, … , 𝑛; 𝑗 = 1, 2, … ., 𝑚 ()

𝑧𝑏

𝑖,𝑗 =𝑧

𝑎

𝑖,𝑗 +(1−𝛽

2)𝑧𝑖,𝑗,𝑖𝑓𝑧

𝑖,𝑗 ≥0,∀𝑖 = 1,2,…,𝑛;𝑗 = 1,2,….,𝑚 ()



𝑧𝑏

𝑖,𝑗 −

𝑛



𝑖=1

𝑧𝑏

𝑖,𝑗𝑛1

≤𝜀2, ∀𝑖 = 1, 2, … , 𝑛; 𝑗 = 1, 2, … ., 𝑚 ()

𝑧𝑖,𝑗 ≤𝐶2, ∀𝑖 = 1, 2, … , 𝑛; 𝑗 = 1, 2, … ., 𝑚 ()

The rebalancing process of the second-level balance can be described as Equations ()–(), where 𝑧𝑖,𝑗 is the transmis-

sion performance of component 𝑗in module 𝑖,and𝑧𝑏

𝑖,𝑗 and 𝑧𝑎

𝑖,𝑗 are performance of component 𝑗in module 𝑖before and after

the rebalancing operation between in-module components, respectively. Before the second-level balance, 𝑧𝑏

𝑖,𝑗 =𝑦

𝑎

𝑖,𝑗 .To

ensure that the input performance equals the output performance among components in a module, Equation ()

expresses this principle. For components that output performance, Equation () represents its resulting performance

by subtracting the output performance from its original performance. Equation () applies to components whose input

performance, with transmission loss, is accounted for by adding the original performance and input performance. 𝛽2rep-

resents the transmission loss between components. When the performance differences between components fall below

𝜀2, the second-level balance is completed, as indicated in Equation (). Furthermore, Equation () specifies that the

component performance output must not exceed 𝐶2.

Finally, according to the definition of system work, the system reliability can be expressed as

𝑅(𝑡) = Pr{𝑆(𝑡) ≥𝐷,𝜎1(𝑡) ≤𝜀1,𝜎

2(𝑡) ≤𝜀2}()

TIAN  . 9

3SYSTEM RELIABILITY EVALUATION CONSIDERING EPISTEMIC UNCERTAINTY

This section proposes a quantification process of epistemic uncertainty based on the DSET. Then, the system reliability

assessment of TLBS-CBPS is proposed based on the BUGF method. Furthermore, to avoid combinatorial explosion, a

simplified evaluation algorithm is given.

3.1 Epistemic uncertainty quantification based on DSET

Due to the epistemic uncertainty caused by the estimation errors of the component’s performance, the performance state of

component jin module iis considered likely to be a set, [𝑔]𝑖,𝑗 ={[𝑔]

𝑖,𝑗,1,[𝑔]

𝑖,𝑗,2,…,[𝑔]

𝑖,𝑗,𝐻}, where [𝑔]𝑖,𝑗,ℎ, ℎ = 1,2,…,𝐻,

maybe a value or a set.

In addition, the uncertainty in a state is described by a probability interval, with the probability interval sets of the state

[𝑔]𝑖,𝑗,ℎ expressed as [p

¯𝑖,𝑗,ℎ(𝑡), 

𝑝𝑖,𝑗,ℎ(𝑡)]. The performance of batteries may vary due to charge–discharge cycles and self-

degradation, which can cause them to operate at different levels. We assume that the performance levels of components

follow a continuous-time discrete-state Markov stochastic process. The differential equations presented below can derive

the upper and lower limits of the probability, [𝜆ℎ2,ℎ1,

𝜆ℎ2,ℎ1]and [𝜇ℎ1,ℎ2

,

𝜇ℎ1,ℎ2]are the upper and lower limits of transition

intensities between performance level ℎ1and level ℎ2,ℎ2>ℎ

1. More specifically, 𝜆is the transition intensity from high

to low performance level, and 𝜇is the transition intensity from low to high performance level. The initial conditions are

¯𝑖,𝑗,𝐻(0) = 1, 

𝑝𝑖,𝑗,𝐻(0) = 1,p

¯𝑖,𝑗,ℎ(0) = 0, 

𝑝𝑖,𝑗,ℎ(0) = 0, ℎ = 1, 2, … , 𝐻 − 1.











𝑑p

¯𝑖,𝑗,1(𝑡)

𝑑𝑡 =

𝐻



ℎ=2

¯𝑖,𝑗,ℎ(𝑡)𝜆ℎ,1 −p

¯𝑖,𝑗,1(𝑡)

𝐻



ℎ=2



𝜇1,ℎ

𝑑p

¯𝑖,𝑗,𝑤(𝑡)

𝑑𝑡 =

𝑤−1



𝐻=1

¯𝑖,𝑗,ℎ(𝑡)𝜇ℎ,𝑤 +

𝐻



ℎ=𝑤+1

¯𝑖,𝑗,ℎ(𝑡)𝜆ℎ,𝑤 −p

¯𝑖,𝑗,𝑤(𝑡) 𝑤−1



ℎ=1



𝜆𝑤,ℎ +

𝐻



ℎ=𝑤+1



𝜇𝑤,ℎ,𝑤 = 2,…,𝐻 −1

𝑑p

¯𝑖,𝑗,𝐻(𝑡)

𝑑𝑡 =

𝐻−1



ℎ=1

¯𝑖,𝑗,ℎ(𝑡)𝜇ℎ,𝐻 −p

¯𝑖,𝑗,𝐻(𝑡)

𝐻−1



ℎ=1



𝜆𝐻,ℎ

,()











𝑑

𝑝𝑖,𝑗,1(𝑡)

𝑑𝑡 =

𝐻



ℎ=2



𝑝𝑖,𝑗,ℎ(𝑡) 

𝜆ℎ,1 −

𝑝𝑖,𝑗,1(𝑡)

𝐻



ℎ=2

𝜇1,ℎ

𝑑

𝑝𝑖,𝑗,𝑤(𝑡)

𝑑𝑡 =

𝑤−1



𝐻=1



𝑝𝑖,𝑗,ℎ(𝑡) 

𝜇ℎ,𝑤 +

𝐻



ℎ=𝑤+1



𝑝𝑖,𝑗,ℎ(𝑡) 

𝜆ℎ,𝑤 −

𝑝𝑖,𝑗,𝑤(𝑡) 

𝑤−1



ℎ=1

𝜆

𝑤,ℎ

𝐻



𝑗=𝑤+1

𝜇𝑤,ℎ

,𝑤 = 2,…,𝐻 −1

𝑑

𝑝𝑖,𝑗,𝐻(𝑡)

𝑑𝑡 =

𝐻−1



ℎ=1



𝑝𝑖,𝑗,ℎ(𝑡) 

𝜇ℎ,𝐻 −

𝑝𝑖,𝑗,𝐻(𝑡)

𝐻−1



ℎ=1

𝜆𝐻,ℎ

()

Then, 𝐩𝑖,𝑗 (𝑡)=[[p

¯𝑖,𝑗,1(𝑡), 

𝑝𝑖,𝑗,1(𝑡)], [p

¯𝑖,𝑗,2(𝑡), 

𝑝𝑖,𝑗,2(𝑡)], … , [p

¯𝑖,𝑗,𝐻(𝑡), 

𝑝𝑖,𝑗,𝐻(𝑡)]], the set of performance state probabilities

of components at time tcan be obtained.

To quantify epistemic uncertainty, the DSET is utilized, which employs the basic probability assignment (BPA) to

describe uncertain information and belief (or support) and plausibility functions to provide upper and lower bounds.

The BPAt is commonly referred to as a mass function, 𝑚(𝑋).The𝑚(𝑋) is specific to the set 𝑋, and each of the subsets has

its mass. From the mass assignments, the upper and lower bounds of the probability interval can be defined by belief and

plausibility functions. Further details can be found in the references., 𝐵𝑒𝑙(𝑋) is defined as the sum of all the masses of

subsets of set 𝑋,while𝑃𝑙(𝑋) is defined as the sum of all the masses of the sets that intersect set 𝑋. They are mathematically

expressed as

𝐵𝑒𝑙(𝑋) = 

𝑌⊆𝑋

𝑚(𝑌), ()

10 TIAN  .

where 𝑌is a subset of 𝑋.

𝑃𝑙(𝑋) = 

𝑌∩𝑋≠𝜙

𝑚(𝑌) ()

where 𝑌is a set that intersects 𝑋.

𝐵𝑒𝑙(𝑋) represents the total quality of information that fully supports the occurrence of event 𝑋, and hence describes

the degree of confidence that event 𝑋is true. It is often regarded as the lower limit of the probability of event 𝑋. On the

other hand, 𝑃𝑙(𝑋) represents the total quality of information that fully or partially supports the occurrence of event 𝑋,

and hence describes the nonfalse confidence of event 𝑋. It is often regarded as the upper limit of the probability of event

𝑋occurring.

Cheng et al. proposed an efficient approximate method to get state mass function based on probability intervals as

follows:

𝑚𝑖,𝑗(𝐸, 𝑡 ) = 









¯𝑖,𝑗,ℎ(𝑡), 𝐸 = [𝑔]𝑖,𝑗,ℎ,ℎ = 1,2,…,𝐻.

1−𝐻

ℎ=1 p

¯𝑖,𝑗,ℎ(𝑡), 𝐸 = [𝑔]𝑖,𝑗,1,[𝑔]

𝑖,𝑗,2,…,[𝑔]

𝑖,𝑗,𝐻

0, 𝑒𝑙𝑠𝑒

,()

where 𝑚𝑖,𝑗(𝐸, 𝑡) is basic belief assignment (BBA, or mass function) of the state space [𝑔]𝑖,𝑗 for component 𝑗in module 𝑖.

The set 𝐸comprises focal elements, which represent all state sets of components 𝑗. It is noteworthy that the number of

elements in set 𝐸is different from the number of elements in the original state space.

After the mapping of 𝑚𝑖,𝑗(𝐸, 𝑡 ), the focal element set of component 𝑗in module 𝑖is {𝐠}𝑖,𝑗 =

{[g]𝑖,𝑗,1,[g]

𝑖,𝑗,2,…, [g]

𝑖,𝑗,𝐻,{[g]

𝑖,𝑗,1,[g]

𝑖,𝑗,2,…, [g]

𝑖,𝑗,𝐻}} = {{g}𝑖,𝑗,1, {g}𝑖,𝑗,2, … , {g}𝑖,𝑗,𝐻1}, where 𝐻1= 𝐻 + 1. {g}𝑖,𝑗,𝐻1includes

all possible subdivision performance values of the component, and the cardinality of this set is denoted as 𝐻2.As

an illustration, if the performance state level of component 𝑗in module 𝑖is {[0],[1, 2],[3, 4]}, the focal element set of

components 𝑗will comprise {{0},{1, 2},{3, 4},{0, 1, 2, 3, 4}}, where 𝐻1=and𝐻2=. {0,1,2,3,4} includes all possible

subdivision performance values of the component. Subsequently, the mass function set associated with {𝐠}𝑖,𝑗 can be

obtained as 𝑚𝑖,𝑗 (𝑡) = {𝑚𝑖,𝑗,1(𝑡), 𝑚𝑖,𝑗,2(𝑡), … , 𝑚𝑖,𝑗,𝐻1(𝑡)}, which represents the degree of confidence for each focal element.

3.2 System reliability evaluation using the BUGF method

The BUGF is a method for addressing epistemic uncertainty within the framework of evidence theory, which extends the

UGF method. The u-function of the performance level of component jin module ican be given,

𝑢𝑖,𝑗(𝑧, 𝑡) =

𝐻1



ℎ=1

𝑚𝑖,𝑗,ℎ(𝑡)𝑧{𝑔}𝑖,𝑗,ℎ ()

For series systems, the recursive algorithm presented below can be employed to obtain the u-function of the

performance levels of all components in the system, denoted as 𝑈𝑛(𝑧, 𝑡). The algorithm is as follows:

First, assign 𝑈𝑖,0 (𝑧, 𝑡) = 𝑧{} . Second, for any 𝑖;for𝑗=,,...,𝑚, repeat:

𝑈𝑖,𝑗(𝑧,𝑡)=𝑈

𝑖,𝑗−1(𝑧, 𝑡) ⊗

∗𝑢𝑖,𝑗(𝑧, 𝑡)

𝑆𝑖,𝑗−1



𝑠𝑖,𝑗−1=1

𝜔𝑖,𝑗−1,𝑠𝑖,𝑗−1 (𝑡)𝑧{𝐠}𝑖,𝑗−1,𝑠𝑖,𝑗−1 ⊗

∗

𝐻1



ℎ=1

𝑚𝑖,𝑗,ℎ(𝑡)𝑧{𝑔}𝑖,𝑗,ℎ

𝑆𝑖,𝑗−1



𝑠𝑖,𝑗−1=1

𝐻1



ℎ=1

𝜔𝑖,𝑗−1,𝑠𝑗−1(𝑡)𝑚𝑖,𝑗,ℎ(𝑡)𝑧{𝐠}𝑖,𝑗−1,𝑠𝑗−1,{𝑔}𝑖,𝑗,ℎ

𝑆𝑖,𝑗



𝑠𝑖,𝑗=1

𝜔𝑖,𝑗,𝑠𝑖,𝑗 (𝑡)𝑧{𝐠}𝑖,𝑗,𝑠𝑖,𝑗 =

𝑆𝑖,𝑗



𝑠𝑖,𝑗=1

𝜔𝑖,𝑗,𝑠𝑖,𝑗 (𝑡)𝑧{𝑔}𝑖,1,ℎ,…,{𝑔}𝑖,𝑗,ℎ𝑖,𝑗,𝑠𝑖,𝑗 .

()

𝑈𝑖,𝑚 (𝑧, 𝑡) = 𝑆𝑖,𝑚

𝑠𝑖,𝑚=1 𝛼𝑖,𝑚,𝑠𝑖,𝑚 (𝑡)𝑧{{𝑔}}𝑖,𝑚,𝑠𝑖,𝑚 can be obtained. Let 𝑈𝑖,𝑚 (𝑧, 𝑡) = 𝑢𝑖(𝑧, 𝑡).

TIAN  . 11

Then, assign 𝑈0(𝑧, 𝑡) = 𝑧{} .For𝑖=,,...,𝑛, repeat:

𝑈𝑖(𝑧, 𝑡) = 𝑈𝑖−1(𝑧, 𝑡) ⊗

∗𝑢𝑖(𝑧, 𝑡)

𝑆𝑖−1



𝑠𝑖−1=1

𝛾𝑖,𝑠𝑖−1(𝑡)𝑧{{𝐠}}𝑖−1,𝑠𝑖−1 ⊗

∗

𝑆𝑖,𝑚



𝑠𝑖,𝑚=1

𝜔𝑖,𝑚,𝑠𝑖,𝑚(𝑡)𝑧{𝐠}𝑖,𝑚,𝑠𝑖,𝑚

𝑆𝑖−1



𝑠𝑖−1=1

𝑆𝑖,𝑚



𝑠𝑖,𝑚=1

𝛾𝑖,𝑠𝑖−1(𝑡)𝜔𝑖,𝑚,𝑠𝑖,𝑚(𝑡)𝑧{{𝐠}}𝑖−1,𝑠𝑖−1,{𝐠}𝑖,𝑚,𝑠𝑖,𝑚

𝑆𝑖



𝑠𝑖=1

𝛾𝑖,𝑠𝑖(𝑡)𝑧{{𝐠}}𝑖,𝑠𝑖=

𝑆𝑖



𝑠𝑖=1

𝛾𝑖,𝑠𝑖(𝑡)𝑧{𝐠}1,𝑚,𝑠1,𝑚 ,…,{𝐠}𝑖,𝑚,𝑠𝑖,𝑚 𝑖,𝑠𝑖.

()

Finally, 𝑈𝑛(𝑧, 𝑡) = 𝑆𝑛

𝑠𝑛=1 𝛾𝑛,𝑠𝑛(𝑡)𝑧{{𝐠}}𝑛,𝑠𝑛can be given.

In Equation (), {𝑔}𝑖,𝑗,𝑠𝑖,𝑗 is a set consisting of the focal elements (performance states) of the first 𝑗components in

module 𝑖under realization 𝑠𝑖,𝑗,and𝜔𝑖,𝑗,𝑠𝑖,𝑗(𝑡) is corresponding probability. The term 𝑆𝑖,𝑗 refers to the total number of

possible realizations of {𝑔}𝑖,𝑗,𝑠𝑖,𝑗 . The operator ⊗

∗is utilized to compute the first 𝑗elements and the 𝑗th component in

module 𝑖recursively. When a new component 𝑗enters the loop, a new vector {{𝑔}𝑖,𝑗−1,𝑠𝑗−1,{𝑔}

𝑖,𝑗,ℎ}is generated. For example,

0.1𝑧{0} ⊗

∗0.2 𝑧{1,2} = 0.02𝑧{{1,2},{0}}.𝑈𝑖,𝑚(𝑧, 𝑡) is u-function that comprises sets of performance states of all components in

module 𝑖.

Similarly, in Equation (), {{𝑔}}𝑖,𝑠𝑖is a set containing performance states of all components of the first 𝑖modules under

realization 𝑠𝑖,and𝛾𝑖,𝑠𝑖(𝑡) is corresponding probability. The term 𝑆𝑖refers to the total number of possible realizations of

{{𝑔}}𝑖,𝑠𝑖. The operator ⊗

∗is also used to compute the first 𝑖modules’ u-function and the 𝑖th module’s u-function recursively.

For example, 0.1𝑧{{1,2},{0}} ⊗

∗0.2 𝑧{{1,2},{1,2}} = 0.02𝑧{{{1,2},{0}},{{1,2},{1,2}}}.

To screen out realizations that the system works properly, indicative functions are established as follows.

𝐼𝐹1{𝑔1,1(𝑡), 𝑔1,2(𝑡), … , 𝑔1,𝑚(𝑡 )}, … , {𝑔𝑛,1(𝑡), 𝑔𝑛,2(𝑡), … , 𝑔𝑛,𝑚(𝑡)}

=1

𝐠(𝑡)−mean(𝐠(𝑡))

∞≤𝜀2,

()

𝐼𝐹2{𝑔1,1(𝑡), 𝑔1,2(𝑡), … , 𝑔1,𝑚(𝑡 )}, … , {𝑔𝑛,1(𝑡), 𝑔𝑛,2(𝑡), … , 𝑔𝑛,𝑚(𝑡)}

=1min 𝑔1,1 (𝑡), 𝑔1,2 (𝑡), … , 𝑔1,𝑚 (𝑡), … ., 𝑔𝑛,1(𝑡), 𝑔𝑛,2(𝑡), … , 𝑔𝑛,𝑚(𝑡)>𝐷

.

()

The indicative function 𝐼𝐹1is used to judge whether the performance between components is balanced, and 𝐼𝐹2is used

to judge whether the total performance of the system meets the demand.

The two-level rebalancing process is described by functions 𝜑1,𝜑2,and𝜑3as follows.

𝜑1𝑥𝑏

1,𝑥

𝑏

2,…,𝑥

𝑏

𝑛;𝐝;𝛼

=𝑥𝑎

1,𝑥

𝑎

2,…,𝑥

𝑎

𝑛,()

𝜑2𝑦𝑏

1,1,𝑦

𝑏

1,2,…,𝑦

𝑏

1,𝑚,…,𝑦

𝑏

𝑛,1,𝑦

𝑏

𝑛,2,…,𝑦

𝑏

𝑛,𝑚;𝑥

𝑎

1,…,𝑥

𝑎

𝑛;𝑥

𝑏

1,…,𝑥

𝑏

𝑛

=𝑦𝑎

1,1,𝑦

𝑎

1,2,…,𝑦

𝑎

1,𝑚,…,𝑦

𝑎

𝑖,1,𝑦

𝑎

𝑖,2,…,𝑦

𝑎

𝑖,𝑚,…,𝑦

𝑎

𝑛,1,𝑦

𝑎

𝑛,2,…,𝑦

𝑎

𝑛,𝑚,

()

𝜑3𝑧𝑏

1,1,𝑧

𝑏

1,2,…,𝑧

𝑏

1,𝑚,….,𝑧

𝑏

𝑛,1,𝑧

𝑏

𝑛,2,…,𝑧

𝑏

𝑛,𝑚;𝛽

2;𝑛

1

=𝑧𝑎

1,1,𝑧

𝑎

1,2,…,𝑧

𝑎

1,𝑚,….,𝑧

𝑎

𝑖,1,𝑧

𝑎

𝑖,2,…,𝑧

𝑎

𝑖,𝑚,….,𝑧

𝑎

𝑛,1,𝑧

𝑎

𝑛,2,…,𝑧

𝑎

𝑛,𝑚.

()

The operational principles of the three functions (𝜑1,𝜑2,and𝜑3) are embodied in the three mentioned nonlinear pro-

gramming approaches. Taking 𝜑1as an example, input 𝑥𝑏

1,𝑥

𝑏

2,…,𝑥

𝑏

𝑛,𝐝,𝛼to get 𝑥𝑎

1,𝑥

𝑎

2,…,𝑥

𝑎

𝑛by Equations ()–(), where

𝛼is transmission loss rate and 𝐝is distance matrix between modules. 𝑥𝑏

𝑖is the performance of module 𝑖before the first-

level balance, that is, 𝐺𝑖(𝑡) = 𝑠𝑢𝑚(𝐆𝑖(𝑡)). Similarly, the functions 𝜑2,𝜑3are implemented using Equations ()–()and

()–(), respectively. If the above linear programming problem has no solution, a zero vector of the corresponding length

12 TIAN  .

should be outputted. In the expression for BUGF, 𝑦𝑏

𝑖,𝑗 and 𝑧𝑏

𝑖,𝑗 represent the performance states of components 𝑔𝑖,𝑗(𝑡) before

the first- and second-level balance, respectively.

Next, the function 𝜉is constructed to judge whether the system can work properly or fail under each realization. To

apply DSET, it is necessary to extend the focal element sets of the components (the mapped set of component performance

levels) to all possible subdivision realizations (subrealizations) consisting of specific component performance values. For

instance, suppose the focal element sets of three components are {}, {, }, and {, }. In that case, by expanding the sets,

all possible subrealizations can be obtained, such as {,,}, {,,}, {,,}, and {,,}.

In particular, if all components are at the performance level {𝑔}𝑖,𝑗,𝐻1, then the expanded realization should contain

𝐻𝑛1

2possible subrealizations. Taking this special case as an example, we introduce the function 𝜉by Equation (a,b,c).

Equation (a) is to carry out the 𝑛1-permutations with repetition over the focal elements set, resulting in a total of 𝐻𝑛1

possible subrealizations. The second step, as described in Equation (b), is to judge whether the system works or fails

under each realization. Equation (c) means that if the system works, it is W; otherwise, it is F. The specific condition of

judgment is given in Equation (), where 𝐼specifically expands to Equation ().

𝜉{𝐠}1,1, {𝐠}1,2 ,…,{𝐠}

1,𝑚,…,{𝐠}

𝑛,1, {𝐠}𝑛,2,…,{𝐠}

𝑛,𝑚

=𝜉

𝑔1,1,1,𝑔

1,2,1,…,𝑔

1,𝑚,1,…,𝑔

𝑛,1,1,𝑔

𝑛,2,1,…,𝑔

𝑛,𝑚,1,

𝑔1,1,2,𝑔

1,2,1,…,𝑔

1,𝑚,1,…,𝑔

𝑛,1,1,𝑔

𝑛,2,1,…,𝑔

𝑛,𝑚,1,

𝑔1,1,1,𝑔

1,2,2,…,𝑔

1,𝑚,1,…,𝑔

𝑛,1,1,𝑔

𝑛,2,1,…,𝑔

𝑛,𝑚,1,…,

𝑔1,1,𝐻2,𝑔

1,2,𝐻2,…,𝑔

1,𝑚,𝐻2,…,𝑔

𝑛,1,𝐻2,𝑔

𝑛,2,𝐻2,…,𝑔

𝑛,𝑚,𝐻2



(a)

=𝜉𝐠(𝑡)1,𝐠(𝑡)

2,…,𝐠(𝑡)

𝐻𝑛1

2(b)

={𝐖, 𝐅},(c)

𝜉{𝐠(𝑡)}=𝑊𝑖𝑓𝐼

𝐹1(𝐠(𝑡))⋅𝐼

𝐹2(𝐠(𝑡))+1−𝐼

𝐹1(𝐠(𝑡))⋅{𝐼}=1,

𝐹𝑒𝑙𝑠𝑒, ()

𝐼=𝐼

𝐹1[𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼

),𝐠(𝑡)

)] ⋅𝐼

𝐹2[𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼

),𝐠(𝑡)

)]

+1−𝐼

𝐹1[𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼

),𝐠(𝑡)

)]⋅𝐼

𝐹1[𝜑3(𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼

),𝐠(𝑡)

);𝛽

2;𝑛

1)] ⋅

𝐼𝐹2[𝜑3(𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼

),𝐠(𝑡)

);𝛽

2;𝑛

1)]

()

Equation () represents a decision-making process to determine whether the system is reliable or not. When the

performance differences between components within the system are below 𝜀2and the system performance meets the

requirement, the system can work, and the term “𝐼𝐹1(𝑔(𝑡)) ⋅ 𝐼𝐹2(𝑔(𝑡))” is used to represent this scenario.

When the system is unbalanced, the value of 1−𝐼

𝐹1(𝑔(𝑡)) is equal to , indicating that the system needs to perform a two-

level rebalancing operation. The first level is to balance the performance of modules, and if the performance difference

between components is reduced to within 𝜀2and the system’s performance meets the demand after the first-level bal-

ance, the system can work, as represented by 𝐼𝐹1(𝜑2(𝜑1(𝐆(𝑡), 𝐝, 𝛼), 𝑔(𝑡))) ⋅ 𝐼𝐹2(𝜑2(𝜑1(𝐆(𝑡), 𝐝, 𝛼), 𝑔(𝑡))) = . Otherwise,

the second-level balance needs to perform. If the system can operate normally after the two-level rebalancing pro-

cess, 𝐼𝐹1(𝜑3(𝜑2(𝜑1(𝐆(𝑡), 𝐝, 𝛼), 𝑔(𝑡)); 𝛽2;𝑛

1)) ⋅ 𝐼𝐹2(𝜑3(𝜑2(𝜑1(𝐆(𝑡),𝐝,𝛼), (𝑡)); 𝛽2;𝑛

1)) = 1. Therefore, 𝐼𝐹1(𝑔(𝑡)) ⋅ 𝐼𝐹2(𝑔(𝑡)) +

[1 − 𝐼𝐹1(𝑔(𝑡))] ⋅ {𝐼} =  means that the system can work normally.

Then, the belief function and plausibility function under epistemic uncertainty at any time tcan be represented as

𝐵𝑒𝑙(𝑊, 𝑡) = 𝑈𝑛(𝑧, 𝑡)𝛿−

𝑊{{𝐠}}𝑛,𝑠𝑛,()

and

𝑃𝑙(𝑊,𝑡)=𝑈

𝑛(𝑧, 𝑡)𝛿+

𝑊{{𝐠}}𝑛,𝑠𝑛,()

where

TIAN  . 13

FIGURE 5 The schematic diagram of the simplified BUGF(S-BUGF) algorithm. BUGF, belief universal generating function.

𝛿−

𝑊{{𝐠}}𝑛,𝑠𝑛=









1, 𝑖𝑓𝜉 {{𝐠}}𝑛,𝑠𝑛⊆𝑊

0, 𝑒𝑙𝑠𝑒

,()

𝛿+

𝑊{{𝐠}}𝑛,𝑠𝑛=









1, 𝑖𝑓𝜉 {{𝐠}}𝑛,𝑠𝑛∩𝑊 ≠𝜙

0, 𝑒𝑙𝑠𝑒

.()

The uncertainty interval of system reliability is

𝑅 ∈ [𝐵𝑒𝑙(𝑊, 𝑡), 𝑃 𝑙(𝑊, 𝑡)].

3.3 Simplified evaluation algorithm of the system reliability

Although the aforementioned method solves the reliability assessment problem, it suffers from the issue of combinatorial

explosion, which can significantly increase computation time. Therefore, we present a simplified algorithm for BUGF to

improve the computation efficiency.

The BUGF method proposed in Section . considers the order of components in the system. However, in a series

system, swapping the position of components within a module does not affect system reliability. For instance, a module

with three component states of {,,} or {,,} undergoes the same second-level rebalancing process regardless of their

order. Similarly, swapping the position of modules symmetrically does not affect the system reliability, even though the

transmission loss rate between modules depends on the distance. For example, when three module performance in the

system is {,,} or {,,}, both scenarios perform the same first-level rebalancing process. Hence, in the computation of

the above BUGF method, significant calculation simplification can be achieved by merging realizations.

Next, the following describes the execution process of the simplified evaluation algorithm, named the S-BUGF algo-

rithm, and its schematic diagram is given in Figure . The main steps of the algorithm include merging, judging, labeling,

expansion, and calculation. To enhance the understanding of Figure , we provide a detailed explanation of it as follows.

Step 1: Merge realizations.

Substep .. Merge the realizations with epistemic uncertainty.

Specifically, in a system with three modules, each containing two components, merge realizations with the same per-

formance level set of components, disregarding the internal component’s location within the module (as demonstrated in

the orange dotted box depicted in Figure ). Then, the realizations that have the same module performance level after a

symmetrical arrangement of modules can be merged (as illustrated in the red dotted box portrayed in Figure ).

14 TIAN  .

FIGURE 6 The schematic diagram of the merging process for realizations with epistemic uncertainty.

Substep .. Expand each realization after merging to all possible subrealizations, then repeat substep . for them.

Due to the consideration of the epistemic uncertainty, a component may have multiple performance values when it is

at a certain performance level. According to Equation (), the mapped performance level {𝑔}𝑖,𝑗,𝐻1includes all possible

performance values of the component. By extending the realization where all components are at a performance level

containing all possible performance values, all possible subrealizations can be obtained.

For example, consider three component performance levels: {}, {}, {, }, assuming all three components are at per-

formance level {, }. Then, we can expand them to obtain the following realizations: {, , }, {, , }, {, , }, {, ,

}, {, , }, {, , }, {, , }, and {, , }. When three components are at performance level {} or {}, the subrealizations

obtained by expanding are included in the above realizations. Therefore, by generating all possible permutations of com-

ponent performance values, all possible subrealizations are obtained. Then, following the merging rule of substep ., the

subrealizations obtained are merged to decrease the number of realizations.

Step 2: Judge whether the system works under each subrealization.

Based on the working mode of the system, all possible subrealizations are judged whether the system works or fails, as

showninthefirstpartofFigure. Then, the operational information of the system under all possible subrealizations is

stored. The remaining subrealizations obtained in substep . can be matched with the existing information to obtain the

corresponding system working states.

Step 3: Label the merged realizations.

As shown in the third and fourth parts shown in Figure , the labels “Bel” or “Pl” can be assigned to the realizations

with epistemic uncertainty based on the system working and failure conditions of their subrealizations. If there exists at

least one subrealization that the system can work, it is labeled as “Pl”; if the system can work under all subrealizations, it

is labeled as “Bel”.

Step 4: Expand the merged realizations and perform the BUGF calculation.

In this step, the merged realizations obtained in substep . are expanded in the reverse direction. Based on the perfor-

mance probabilities of components and the labels assigned to the realizations in Step , the belief function and plausibility

function performance distributions of the entire system are determined. This information is then used to calculate the

uncertainty interval of the system reliability.

TIAN  . 15

FIGURE 7 The schematic diagram of the process of judging and labeling for realizations.

FIGURE 8 State transition intervals of components in the small UAV energy storage system with epistemic uncertainty.

4CASE STUDY

This section presents an analytical case study and a numerical case study. The analytical example provides a detailed expla-

nation of the calculation process of the BUGF method. The numerical example further demonstrates the computational

efficiency of the S-BUGF method. Finally, sensitivity analysis is conducted to discuss the impact of different parameters

on the system reliability.

4.1 Analytical example

Taking a small UAV energy storage system as an example, we give the specific calculation process of the uncertainty inter-

val of the system reliability. The system consists of four components, where each module is connected to two components

via a common bus (𝑛 = 2, 𝑚 = 2, 𝑛1=4), and two modules are also connected by a common bus. Due to the absence

of actual statistical data, artificial data have been utilized. The four components follow the Markov process as shown

in Figure . In addition, we set the original parameters as 𝐶1=2, 𝐶

2=1,𝛼=0.02,𝛽

1= 0.01, 𝜀1= 0.0015, 𝜀2=

0.03, 𝐷 = 1.8.

By applying the aforementioned conditions, we can solve Equations ()and() to determine the upper and lower

limits of probability intervals that correspond to the states of each component at time t. The resulting expressions are as

follows:

¯1,1,1(𝑡) = p

¯1,1,[0,1](𝑡) = 0.67082𝑒−0.00003𝑡 − 0.67082𝑒−0.00029𝑡

¯1,1,2(𝑡) = p

¯1,1,[2.5](𝑡) = 0.27639𝑒−0.00003𝑡 + 0.72361𝑒−0.00029𝑡

16 TIAN  .

TABLE 1 Probability intervals of components’ performance level at t=.

Performance level Component 1 in Module 1 Component 2 in Module 1 Component 1 in Module 2 Component 2 in Module 2

[, ] [. .] [. .] [. .] [. .]

[.] [. .] [. .] [. .] [. .]

TABLE 2 BBAs of components at t=.

State {0, 1} {2.5} {0, 1, 2.5}

Component  in Module  . . .

Component  in Module  . . .

Component  in Module  . . .

Component  in Module  . . .

¯1,2,1(𝑡) = −0.61538𝑒−0.00028𝑡 + 0.61538𝑒−0.00002𝑡

¯1,2,2(𝑡) = 0.69231𝑒−0.00028𝑡 + 0.30769𝑒−0.00002𝑡

¯2,1,1(𝑡) = 0.34770𝑒−0.00003𝑡 − 0.60921𝑒−0.00032𝑡

¯2,1,2(𝑡) = 0.60921𝑒−0.00003𝑡 + 0.65230𝑒−0.00032𝑡

¯2,2,1(𝑡) = 0.63059𝑒−0.00003𝑡 − 0.63059𝑒−0.00024𝑡

¯2,2,2(𝑡) = 0.42724𝑒−0.00003𝑡 + 0.57276𝑒−0.00024𝑡



𝑝1,1,1(𝑡) = 0.70276𝑒−0.00003𝑡 − 0.70276𝑒−0.00029𝑡



𝑝1,1,2(𝑡) = 0.34028𝑒−0.00003𝑡 + 0.65972𝑒−0.00029𝑡



𝑝1,2,1(𝑡) = 0.68640𝑒−0.00002𝑡 − 0.68640𝑒−0.00027𝑡



𝑝1,2,2(𝑡) = 0.37988𝑒−0.00002𝑡 + 0.62012𝑒−0.00027𝑡



𝑝2,1,1(𝑡) = 0.63695𝑒−0.00003𝑡 − 0.63695𝑒−0.00032𝑡



𝑝2,1,2(𝑡) = 0.39867𝑒−0.00003𝑡 + 0.60133𝑒−0.00032𝑡



𝑝2,2,1(𝑡) = 0.54956𝑒−0.00003𝑡 − 0.54956𝑒−0.00024𝑡



𝑝2,2,2(𝑡) = 0.40841𝑒−0.00003𝑡 + 0.59159𝑒−0.00024𝑡

Table presents the probability intervals for the states at t=. These probability intervals, along with Equation (),

are then utilized to calculate the state sets and corresponding BBAs, as shown in Table .

Then, based on the method proposed in Section ., the u-function of each component at t= can be given,

𝑢1,1(𝑧) = 0.0831𝑧{0,1} + 0.8975𝑧{2.5} + 0.0194𝑧{0,1,2.5} ,

TIAN  . 17

𝑢1,2(𝑧) = 0.0743𝑧{0,1} + 0.9065𝑧{2.5} + 0.0192𝑧{0,1,2.5} ,

𝑢2,1(𝑧) = 0.0825𝑧{0,1} + 0.8981𝑧{2.5} + 0.0194𝑧{0,1,2.5} ,

𝑢2,2(𝑧) = 0.0608𝑧{0,1} + 0.9290𝑧{2.5} + 0.0103𝑧{0,1,2.5} ,

Based on the recursive algorithm given by Equations ()and(), the u-function of the system can be obtained.

𝑈1,1(𝑧) = 𝑈1,0 (𝑧) ⊗

∗𝑢1,1(𝑧) = 𝑧{} ⊗

∗𝑢1,1(𝑧, 𝑡) = 0.0831𝑧{0,1} + 0.8975𝑧{2.5} + 0.0194𝑧{0,1,2.5} ,

𝑈1,2(𝑧) = 𝑈1,1 (𝑧) ⊗

∗𝑢1,2(𝑧)

= 0.0062𝑧{{0,1},{0,1}}+0.0753𝑧{{0,1},{2.5}}+ 0.0667𝑧{{2.5},{0,1}}

+ 0.0016𝑧{{0,1},{0,1,2.5}}+0.0014𝑧{{0,1,2.5},{0,1}}+0.8136𝑧{{2.5},{2.5}}

+ 0.0172𝑧{{2.5},{0,1,2.5}}+0.0176𝑧{{0,1,2.5},{2.5}}+0.00037𝑧{{0,1,2.5},{0,1,2.5}},

𝑈2,1(𝑧) = 𝑈2,0 (𝑧) ⊗

∗𝑢2,1(𝑧) = 𝑧{} ⊗

∗𝑢2,1(𝑧, 𝑡) = 0.0825𝑧{0,1} + 0.8981𝑧{2.5} + 0.0194𝑧{0,1,2.5} ,

𝑈2,2(𝑧) = 𝑈2,1 (𝑧) ⊗

∗𝑢2,2(𝑧)

= 0.0050𝑧{{0,1},{0,1}}+0.0766𝑧{{0,1},{2.5}}+ 0.0546𝑧{{2.5},{0,1}}

+ 0.00085𝑧{{0,1},{0,1,2.5}}+0.0012𝑧{{0,1,2.5},{0,1}}+0.8343𝑧{{2.5},{2.5}}

+ 0.0093𝑧{{2.5},{0,1,2.5}}+0.0180𝑧{{0,1,2.5},{2.5}}+0.0002𝑧{{0,1,2.5},{0,1,2.5}},

where 𝑈2(𝑧) includes  terms. Next, we provide a detailed calculation process for the term {{0, 1}, {2.5}, {2.5}, {0, 1, 2.5}}

by applying Equation ().

𝜉({0, 1}, {2.5}, {2.5}, {0, 1, 2.5})

=𝜉{0, 2.5, 2.5, 0},{1, 2.5, 2.5, 0},{0, 2.5, 2.5, 1},

{1, 2.5, 2.5, 1},{0, 2.5, 2.5, 2.5},{1, 2.5, 2.5, 2.5}

={𝐹,𝐹, 𝐹,𝐹, 𝑊, 𝑊}.

()

Taking the two subrealizations {1,2.5,2.5,0}and {0,2.5,2.5,2.5}as examples, which are the subrealizations of the system

working and failure after rebalancing in Equation (), the detailed calculation procedure is given as follows.

First, by Equations ()and(), we check whether the current system is balanced and meets the demand.

𝐼𝐹1(1({1, 2.5, 2.5, 0})) =1{1, 2.5, 2.5, 0}−mean({1, 2.5, 2.5, 0})∞≤0.03

=1({0.5, 1, 1, 1.5}≤0.03)=0,

𝐼𝐹2({1, 2.5, 2.5, 0}) =1(min ({1, 2.5, 2.5, 0}) >1.8

)=0,

Due to 𝐼𝐹2({1, 2.5, 2.5, 0}) ⋅ 𝐼𝐹3({1, 2.5, 2.5, 0}) = 0, the system needs to perform the rebalancing operation. Subse-

quently, we use Equation () to implement performance rebalancing among modules (the first-level balance) and use

Equation () to obtain the performance of each component after the first-level balance.

𝜑1{1 + 2.5, 2.5 + 0},10

,0.02

={3.0253, 2.9653},

𝜑2({3.0253, 2.9653},{1, 2.5, 2.5, 0}) ={1, 2.0253, 2.5, 0.4653}

Then, a system balance check should be performed to evaluate the current state.

18 TIAN  .

𝐼𝐹1({1, 2.0253, 2.5, 0.4653}) = 0.

The system is still not in compliance with the balance requirement, thus it is necessary to perform the second-level

balance by using Equation (). However, due to the transmission capacity limitation, the system cannot restore balance,

which leads to Equations ()and().

𝜑3({1, 2.0253, 2.5, 0.4653}; 0.01; 4)={0, 0, 0, 0}()

𝐼𝐹2(𝜑3({1, 2.0253, 2.5, 0.4653}; 0.01; 4)) =0 ()

Thus, we can have,

𝐼=𝐼

𝐹2𝜑2𝜑1{3.5, 2.5},10

,0.02

,{1, 2.5, 2.5, 0}

×𝐼

𝐹3𝜑2𝜑1{3.5, 2.5},10

,0.02

,{1, 2.5, 2.5, 0}

+1−𝐼

𝐹2𝜑2𝜑1{3.5, 2.5},10

,0.02

,{1, 2.5, 2.5, 0}

×𝐼

𝐹2𝜑3𝜑2𝜑1{3.5, 2.5},10

,0.02

,{1, 2.5, 2.5, 0}; 0.01; 4

×𝐼

𝐹3𝜑3𝜑2𝜑1{3.5, 2.5},10

,0.02

,{1, 2.5, 2.5, 0}; 0.01; 4=0

and,

𝐼𝐹2({1, 2.5, 2.5, 0}) ⋅𝐼

𝐹3({1, 2.5, 2.5, 0}) +1−𝐼

𝐹2({1, 2.5, 2.5, 0})⋅{𝐼}=0

Finally, we can get 𝜉{{1,2.5,2.5,0}} = {𝐹}.

Similarly, for the subrealization {0,2.5,2.5,2.5},wehave,

𝐼𝐹1(1({0, 2.5, 2.5, 2.5})) =1

𝐠(𝑡)−mean(𝐠(𝑡))

∞≤0.03

=1({1.875, 0.6250, 0.6250, 0.625}≤0.03)=0 ,

which means the system needs to perform the rebalancing operation. After performing the first-level balance, we can

obtain,

𝜑1({2.5, 5},[1, 1],0.02

)={3.7225, 3.7525},

𝜑2({3.7225, 3.7525},{0, 2.5, 2.5, 2.5}) ={1.2225, 2.5, 1.8763, 1.8763},

Due to 𝐼𝐹2({1.2225, 2.5, 1.8763, 1.8763}) = 0, the second-level balance still needs to perform, that is,

𝜑3({1.2225, 2.5, 1.8763, 1.8763}; 0.01; 4) = {1.8372, 1.8791, 1.8763, 1.8763}.

Based on the above results, we have,

𝐼𝐹2({0, 2.5, 2.5, 2.5}) ⋅𝐼

𝐹3({0, 2.5, 2.5, 2.5}) +1−𝐼

𝐹2({0, 2.5, 2.5, 2.5})⋅{𝐼}=1.

andatthispoint,𝜉{{0,2.5,2.5,2.5}} = {𝑊}. Based on Equations ()and(), we can obtain,

𝛿−

𝑊({0, 1}, {2.5}, {2.5}, {0, 1, 2.5})=0,

𝛿+

𝑊({0, 1}, {2.5}, {2.5}, {0, 1, 2.5})=1.

TIAN  . 19

FIGURE 9 Dynamic trends of the four-component system reliability with epistemic uncertainty (∼ h, ∼ h).

Therefore, the following equations can be obtained:

𝐵𝑒𝑙(𝑊)=𝑈

𝑛(𝑧)𝛿−

𝑊{{𝐠}}𝑛,𝑠𝑛

= 3.097 × 10−5 ×0+4.732×10

−4 ×0

+ ⋯ + 0.0629 × 1 + ⋯ + 6.713 × 10−6 ×0+7.443×10

−8 ×0,

𝑃𝑙(𝑊)=𝑈

𝑛(𝑧)𝛿+

𝑊{{𝐠}}𝑛,𝑠𝑛

= 3.097 × 10−5 ×0+4.732×10

−4 ×0

+ ⋯ + 0.0629 × 1 + ⋯ + 6.713 × 10−6 ×1+7.443×10

−8 ×1,

where the sum of coefficients for the term with 𝛿−

𝑤=, 𝐵𝑒𝑙(𝑊), equals ., and the sum of coefficients for the term

with 𝛿+

𝑤=, 𝑃𝑙(𝑊), equals ..

Finally, the upper and lower bounds of the uncertainty interval of the system reliability can be given as [.,.].

The epistemic uncertainty of system reliability at t= is 𝑃𝑙(𝑤) − 𝐵𝑒𝑙 (𝑤) = ..

Considering the maintenance period of the battery at  h, Figure shows the dynamic trend of the system relia-

bility with epistemic uncertainty over time within ∼ h. Additionally, to further investigate the changing of system

uncertainty over a long time, the trend over a longer period is also presented in Figure .

As shown in Figure , the distance between the upper and lower bounds of the system reliability is gradually increas-

ing, indicating an increasing level of system uncertainty. This phenomenon is attributed to the fact that the system state

becomes more uncertain as time progresses.

4.2 Numerical example

This subsection presents a numerical case study of a lithium-ion battery pack with a SP structure, which verifies the

usability of the proposed method in practical applications. Meanwhile, the calculation results of three methods were

compared, including BUGF, S-BUGF, and Monte Carlo (MC) method.

The SP lithium-ion battery pack, consisting of nine batteries connected in series, is a common energy storage com-

ponent in unmanned aerial vehicles and is regarded as TLBS-CBPS. The component performance is set to have three

levels, that is, [], [.,.], [.]. The system demand performance is .. Table presents the transition probabilities for

the performance levels of the batteries with epistemic uncertainty. The possible ranges of the parameters are provided in

Table .

When 𝐶1=2,𝐶

2= 1, 𝛼 = 0.02, 𝛽1= 0.01, 𝜀1= 0.015, and 𝜀1=0.03, we can use the S-BUGF method to evaluate

the system reliability, and the results are shown in Figure .

20 TIAN  .

TABLE 3 Transition probabilities and performance levels of all batteries with epistemic uncertainty.

Batteries Performance level Transition probability

Electricity

set (kWh)

L L L

L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

 L – [.,.] [.,.] []

L [.,.] –[.,.] [.,.]

L [.,.] [.,.] – .

L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

 L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

 L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

 L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

L – [.,.] [.,.] []

L [.,.] – [.,.] [.,.]

L [.,.] [.,.] – [.]

TABLE 4 The setting of parameters’ values.

Parameters Values Sources

𝐶1,𝐶

2.∼,

𝛼,𝛽2.∼. 

𝜀2∼. 

In Figure , it can be observed that the system reliability gradually decreases and eventually reaches a stable state.

This is because the Markov process tends to stabilize when the system runs for a long time. The actual system reliability

will be between the Bel and Pl, and the reliability interval can help engineers better identify and evaluate risks, thereby

formulating more effective risk management plans.

Let 𝐶1=2,𝐶

2= 1, 𝛼 = 0.02, 𝛽1= 0.01, 𝜀1= 0.015,𝜀2=0.03,and𝐷=.. Next, we present the uncertainty inter-

vals of system reliability for the scenarios where the component performance level sets are {[,],.} (Scenario ) and

{,[.,.],.} (Scenario ), respectively. For the convenience of computation, assume that the state transition intervals

for all components in each scenario are the same, and the specific state transition is shown in Figure . In addition, we

increase the number of modules and the number of components within each module to better demonstrate the effec-

tiveness of S-BUGF method. Meanwhile, to verify the accuracy of the results, calculation results of the MC method are

TIAN  . 21

FIGURE 10 Dynamic trends of the nine-component system reliability with epistemic uncertainty (∼ h, ∼ h).

FIGURE 11 State transition intervals of components under two scenarios.

TABLE 5 Results and calculation time by BUGF, S-BUGF, and MC in Scenario .

nmBUGF(s) RS-BUGF(s) RMC(s) R

  . [.,.] . [.,.] . [.,.]

  . [.,.] . [.,.] . [.,.]

  . [.,.] . [.,.] . [.,.]

  . [.,.] . [.,.] . [.,.]

  / / . [.,.] . [.,.]

  / / . [.,.] . [.,.]

  / / . [.,.] / /

Results with computation time exceeding  h are not displayed.

also provided, which is simulated , times. The calculation results and time (CPU/AMD Ryzen  H . GHz,

RAM/.GB)areshowninTablesand .

In Tables and , it can be seen that the calculation time of S-BUGF is significantly shorter than the other two methods.

Moreover, compared with the MC method, it can confirm the accuracy of the calculation results. To a certain extent, the

S-BUGF can address the reliability evaluation for systems with a higher degree of state subdivision.

4.3 Sensitivity analysis

In this section, sensitivity analysis is conducted to investigate the effect of five parameters (𝐶1,𝐶2,𝛼,𝛽2,𝜀2) on the system

reliability. Specifically, based on the data in Table , we varied each parameter individually. The trend graphs of the system

reliability uncertainty intervals with the variation of the five parameters at t=areshowninFigures–.

The system reliability is influenced by five parameters, as shown in Figures –. The system reliability increases with

the increase of transmission capacity limits between modules and in-module components, as well as the balance degree

threshold, while it decreases with the increase of the transmission loss rate. Figure  presents the trend of the system

22 TIAN  .

TABLE 6 Results and calculation time by BUGF, S-BUGF, and MC in Scenario .

nmBUGF(s) RS-BUGF(s) RMC(s) R

  . [.,.] . [.,.] . [.,.]

  . [.,.] . [.,.] . [.,.]

  . [.,.] . [.,.] . [.,.]

  / / . [.,.] . [.,.]

  / / . [.,.] / /

  / / . [.,.] . [.,.]

  / / . [.,.] / /

Results with computation time exceeding  h are not displayed.

FIGURE 12 The sensitivity analysis of the system reliability on 𝐶1when 𝐶2=. and 𝐶2=.

FIGURE 13 The sensitivity analysis of the system reliability on 𝐶2when 𝐶1=and𝐶1=.

reliability changing with 𝐶2for 𝐶1=and𝐶1=, respectively, and it is observed that the changes in the two cases are not

identical. This indicates that the system reliability is influenced by both 𝐶1and 𝐶2, and the limitation of 𝐶1is the reason

why the system reliability cannot reach . The trend of change in the Bel and Pl is roughly the same as the parameters

change, but their trajectories of change are not the same. For example, in Figure , Pl is more sensitive to the impact of

the module intertransmission loss rate (𝛼) than Bel.

The results also indicate that the transmission capacity limits between modules have the most significant impact on the

system reliability, while the transmission loss rate between modules has a relatively minor impact on system reliability.

Moreover, the system reliability is affected by the joint effect of these parameters. These conclusions provide insights

for optimizing TLBS-CBPS configuration, such as maximizing the capacity of the performance storage devices between

modules.

TIAN  . 23

FIGURE 14 The sensitivity analysis of the system reliability on 𝛼and 𝛽2.

FIGURE 15 The sensitivity analysis of the system reliability on 𝜀2.

5CONCLUSIONS AND FUTURE WORK

In this study, a reliability assessment method is proposed for TLBS-CBPS by considering the two-level balance, epistemic

uncertainty, and transmission loss. First, the reliability model is constructed based on the working mode of TLBS-CBPS.

During the two-level rebalancing process, the performance transmission process among modules, between modules and

components, and among components are formulated as three nonlinear programming problems. Second, the BUGF

method is proposed to evaluate the epistemic uncertainty of the system reliability based on the DSET, which involves

converting the system’s operational conditions and solutions to nonlinear programming problems into indicator func-

tions for computation. To alleviate the problem of combinatorial explosion, a simplified BUGF algorithm is proposed by

merging, judging, labeling, and expanding realizations. The effectiveness of the proposed method is demonstrated through

two examples of lithium-ion battery packs and the correctness of the results is verified by MC simulation. Results of sensi-

tivity analysis show that the transmission capacity between modules has the most significant impact on system reliability,

while the transmission loss rate between modules has a relatively small impact. These results can help improve the design

and configuration of such systems in practice.

This study only considers two-level rebalancing operations in the reliability assessment of PBSs-CBPS, future works

could investigate the reliability model of PBSs-CBPS with multiple levels of rebalancing. Next, redundancy backup strate-

gies are commonly used to improve the reliability of PBSs-CBPS, and the corresponding policies can be further studied.

Finally, the maintenance plan for PBSs-CBPS should consider the impact of the balance degree threshold, as repairing one

component may require the repair of other components. Hence, the optimization of maintenance strategies for PBSs-CBPS

is needed.

24 TIAN  .

NOTATIONS

𝑛number of modules in the system

𝑚number of in-module components

𝑛1number of components in the system

𝜀1balance degree threshold between modules

𝜀2balance degree threshold between components

𝐶1transmission capacity limit between modules

𝐶2transmission capacity limit between in-module components.

𝑔𝑖,𝑗 performance of component jin module i,i=,,...,mj=,,...,n

𝐠performance set of all components in the system

𝐆𝑖performance set of components in module i

𝐆performance set of all modules

𝛿1balance degree between modules

𝛿2balance degree between components

𝑆total performance of the system

𝐷demand performance of the system

𝛼transmission loss rate of each pass through the performance storage device

𝑑𝑘,𝑙 the number of performance storage devices between module kand module l

𝛽1

𝑘,𝑙 transmission loss between module kand module l

𝛽2transmission loss between components

𝑥𝑘,𝑙 transmission performance from module kto module l

𝑥𝑏

𝑘performance of module kbefore the performance rebalancing between modules

𝑥𝑎

𝑘performance of module kafter the performance rebalancing between modules

𝑦𝑏

𝑖,𝑗 performance of component jin module ibefore the first-level balance

𝑦𝑎

𝑖,𝑗 performance of component jin module iafter the first-level balance

𝐲𝑎

𝑖performance set of components in module iafter the first-level balance

Δ𝑖,𝑗 performance of transmission between module iand in-module component j

𝑧𝑖,𝑗 transmission performance of component jin module i.

𝑧𝑏

𝑖,𝑗 performance of component jin module ibefore the second-level balance

𝑧𝑎

𝑖,𝑗 performance of component jin module iafter the second level balance

𝐻the performance level number of components

[𝑔]𝑖,𝑗,ℎ performance interval of component jin module iat level h,ℎ = 1, 2, … , 𝐻

[𝐠]𝑖,𝑗 performance interval set of component jin module i,[𝐠]𝑖,𝑗 ={[𝑔]

𝑖,𝑗,1,[𝑔]

𝑖,𝑗,2,…,[𝑔]

𝑖,𝑗,𝐻},𝑗 = 1, 2, … , 𝑛

𝑚(𝐸) basic belief assignment (BBA, mass function) of the state space [𝑔]𝑖,𝑗

{𝑔}𝑖,𝑗,ℎ focal element of component jin module iat level hafter the mapping of 𝑚(𝐸)

{𝐠}𝑖,𝑗 focal element set of component jin module iat level hafter the mapping of 𝑚(𝐸)

¯𝑖,𝑗,ℎ the lower bound of the probability of component j(in module i)’s performance at level h



𝑝𝑖,𝑗,ℎ the upper bound of the probability of component j(in module i)’s performance at level h

𝐩𝑖,𝑗 probability interval sets of the performance level of component jin module i

𝐵𝑒𝑙(𝑋) belief function, the total quality of information that fully supports the occurrence of event 𝑋

𝑃𝑙(𝑋) plausibility function, the total quality of information that fully supports the occurrence of event 𝑋

𝑅system reliability

⊗

∗operator for computing the first 𝑗elements and the 𝑗th component in module 𝑖, recursively

𝑢𝑖,𝑗(𝑧) u-function of the performance level of component jin module i

𝑈𝑛(𝑧) u-function of the performance levels of all components in the system

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant .

TIAN  . 25

DATA AVAILABILITY STATEMENT

Research data are not shared.

ORCID

Jun Yang https://orcid.org/---

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AUTHOR BIOGRAPHIES

Tianzi Tian is currently a Master at the School of Reliability and Systems Engineering, Beihang University. She

received the B.S. at the School of Management Engineering, Zhengzhou University. Her recent research interests

include reliability modeling, maintenance modeling, important measure, and network reliability.

Jun Yang is currently a Professor in Systems Engineering with the School of Reliability and Systems Engineering,

Beihang University, Beijing, China. He received the B.S. degree in Mathematics and Information Science from Yantai

University, and the Ph.D. degree in Probability and Statistics from the Chinese Academy of Sciences. His scientific

interests include reliability modeling, applied statistics, statistical process control, accelerated test, network reliability,

and transfer learning.

Ning Wang received the B.S. degree from the China University of Geosciences, Beijing, China, in . He is currently

a Ph.D. student at the School of Reliability and Systems Engineering, Beihang University, China. His research interests

include network reliability, wireless sensor network optimization, and natural language processing.

Availability Evaluation and Maintenance Optimization of Balanced Systems Considering State-Dependent Inspection Intervals

Article

Full-text available

May 2024
IEEE T RELIAB

There has been increasing attention to the maintenance optimization of balanced systems in recent years. However, existing studies mostly neglect state-dependent inspection intervals and group maintenance, which inadequately addresses the maintenance challenges of balanced systems. Thus, we propose an availability evaluation and maintenance optimization method for balanced systems considering state-dependent inspection intervals. First, multiple maintenance thresholds are introduced to characterize the maintenance strategy considering preventive maintenance and state-dependent inspection intervals, where the next inspection interval is determined based on the post-maintenance system state. Then, the system availability is evaluated by combining semi-regenerative theory and universal generating functions, where calculations are simplified by merging the same system states. Meanwhile, this study also explores the system availability under group maintenance to better reflect reality. Second, the average maintenance cost per unit of time is calculated using the renewal theory. The optimal maintenance thresholds are given by minimizing the maintenance cost under the constraint of minimum system availability. To improve the optimization efficiency, a tabu list-based two-stage iterative partial optimization algorithm is proposed. Finally, the effectiveness of the proposed method is demonstrated through a numerical example involving a lithium-ion battery pack.

Optimal resource allocation in common bus performance sharing systems with transmission loss

Article

Jun 2024

Liudong Gu

The reliability modeling and optimization of performance sharing systems (PSSs) are of vital importance due to their wide applications. Existing research mainly focuses on evaluating and maximizing the reliability of PSSs. However, in many practical systems, decision‐makers tend to prioritize the average cost of the system over its reliability. This paper studies the resource allocation optimization in common bus PSSs. In such systems, each unit has binary‐state random performance to satisfy the multi‐state random demand. The surplus performance can be shared via a common bus with transmission loss between the common bus and the unit. The performance allocation, performance transmission, and unsupplied demand incur costs. Resource allocation strategies are determined by optimization models considering different objective functions and constraints. Additionally, transmission loss between the common bus and the unit is considered. A genetic algorithm is employed to efficiently find the optimal allocation strategies. Numerical examples prove the effectiveness of the proposed models in improving system reliability and reducing system costs.

A neural network copula function approach for solving joint basic probability assignment in structural reliability analysis

Article

May 2024

Applying evidence theory to structural reliability analysis under epistemic uncertainty, it is necessary to consider the correlation of evidence variables. Among them, solving the joint basic probability assignment (BPA) of the evidence variables is a crucial link. In this study, a solution method of joint BPA based on neural network copula function is proposed. This method is to automatically construct copula function through neural network, which avoids the process of selecting the optimal copula function. Firstly, the neural network copula function is constructed based on the sample set of evidence variables. Then, the expression for solving the joint BPA using the neural network copula function is derived through vectors. Furthermore, the expression is used to map the marginal BPA of evidence variables to joint BPA, thus realizing the solution of joint BPA. Finally, the effectiveness of this method is verified by three examples. The results show that the neural network copula function describes the data distribution better than the optimal copula function selected by the traditional method. In addition, there is actually an error in solving the reliability intervals using the traditional optimal copula function method, whereas the results of this paper's neural network copula function method are more accurate and better for decision making.

Reliability modelling of 𝑘-out-of-𝑛: 𝐹 balanced systems with common bus performance sharing

Article

Apr 2024
RELIAB ENG SYST SAFE

This study examines the reliability of k-out-of-n:F balanced systems incorporating common bus performance sharing. The system comprises m sectors, with each sector containing n components. Each component has multiple performance levels to fulfill random demands. All components are interconnected through a common bus, enabling the surplus performance from certain components to be transmitted to deficient components via the bus. The system remains balanced as long as the difference in the number of functional components between any two sectors does not exceed a threshold d. A sector fails when the number of nonfunctional components reaches k, and the entire system fails if either Kf sectors fail or the system becomes unbalanced, whichever occurs first. Existing studies primarily focus on forcing-down functional components or activatin standby components as the primary means of maintaining balance, which often leads to component wastage. However, performance sharing can also achieve rebalancing in practice. This study contributes by developing reliability models for performance sharing in balanced systems and proposing algorithms to determine the optimal performance sharing policy maximizing system reliability. The universal generating function is utilized to calculate the balance probability and system reliability. Furthermore, a case study involving a wing system in a Solar-powered Unmanned Aerial Vehicle is presented to illustrate the effectiveness of the proposed methodology.

Fast Bayesian Inference of Reparameterized Gamma Process With Random Effects

Article

Full-text available

Jan 2023

In the field of reliability engineering, the gamma process plays an important role in modeling degradation processes. However, extracting lifetime information from product degradation observations has long been suffering from both ineffective modeling techniques and inefficient statistical inference methods. To overcome these challenges, we propose a reparameterized gamma process with random effects in this article. Compared with the classical gamma process, the proposed model has a more intuitive physical interpretation. In addition, statistical inference for the model can be readily done through the variational Bayesian algorithm. Combining with the Gauss–Hermite quadrature and the Laplace approximation, the algorithm yields closed-form variational posteriors for the proposed model. Its superiority over two other inference methods (expectation maximization and Monte Carlo Markov Chain) in terms of computational efficiency and estimation accuracy is demonstrated by simulation.

Reliability assessment of performance-based balanced systems with rebalancing mechanisms

Article

Feb 2023
RELIAB ENG SYST SAFE

Reliability assessment of performance-based balanced systems with rebalancing mechanisms, Reliability Engineering and System Safety (2023), doi: https://doi.

Multi-State Balance System Reliability Research Considering Load Influence

Article

May 2023
RELIAB ENG SYST SAFE

Reliability analysis of consecutive-k-out-of-r-from-n subsystems: F balanced systems with load sharing

Article

Aug 2022
RELIAB ENG SYST SAFE

In this paper, a consecutive-k-out-of-r-from-n subsystems: F balanced system with load sharing is proposed, which takes both linear and circular structures into consideration. Such a system has n subsystems and each subsystem consists of several sectors that have multiple components. Each subsystem should keep balance when it works and the concept of balance means that the number of working components should be the same in any sector belonging to the same subsystem. The system maintains balance by forcing down working components to standby or resuming standby components to operate. The failure or standby of a component will result in a higher load on each remaining working component, thereby inducing a higher hazard rate for them. The whole system fails if it contains a set of r consecutive subsystems in which at least k subsystems break down. The Markov process method and finite Markov chain imbedding approach are applied to analyze the system reliability. Numerical examples are presented to show the effectiveness of the studied model and proposed method.

Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints

Article

Oct 2021
RELIAB ENG SYST SAFE

Modern systems are operating under dynamic environments and the components therein often exhibit positively correlated lifetimes. Moreover, due to various practical reasons such as the load sharing mechanism, it is not uncommon that lifetimes of some components dominate the others within the same system. In this study, we first propose a statistical model for system reliability evaluation by jointly considering the correlated component lifetimes and the lifetime ordering constraints. In specific, the effects of the dynamic environments are incorporated by modelling the cumulative hazard function as an exponential dispersion process and the lifetime ordering constraints are modelled by truncating the support of the joint lifetime distribution. We then discuss the statistical inference based on the proposed model. The point estimates of the model parameters as well as the lifetime quantiles are obtained by the maximum likelihood method, and the confidence intervals are constructed by using the generalized pivots. Extensive simulation show that the proposed interval estimation procedures can achieve accurate coverage even under small sample sizes. Two real examples are used to illustrate the proposed modelling and estimation framework.

Reliability evaluation of consecutive k -out-of- m : F balanced systems with a symmetry line

Article

Aug 2021

Based on some real backgrounds, a new balanced system structure, a consecutive k-out-of- m: F system with a symmetry line, is proposed in this paper. Considering different state numbers of a subsector, the new balanced system is analyzed under two situations respectively: the subsector with binary-state and the subsector with multi-state, while the multi-state balanced systems have not been studied in the previous research. Besides, two models are developed in terms of assumptions for the two situations, respectively. For this system, several methods, such as the finite Markov chain imbedding approach, the order statistics technique and the phase-type distributions, are used on the models. In addition to system reliability formulas, the means and variances of the system lifetimes under two models for different situations are given. Finally, numerical examples are presented to illustrate the results obtained in this paper.

Reliability Assessment of Multi-State Phased Mission Systems With Common Bus Performance Sharing Subjected to Epistemic Uncertainty

Article

May 2021

In many real situations, because of the lack or inaccuracy of data, it is difficult to evaluate the performance levels and state probabilities of multistate components with precise values. Thus, the reliability evaluation of systems is always affected by the epistemic uncertainty. Existing research on epistemic uncertainty just focuses on simple multistate systems, without considering the phased mission and performance sharing characteristics of systems. Besides, the impact of transmission loss and performance storage on reliability needs to be considered in the modeling of performance sharing systems. In this article, considering the epistemic uncertainty, transmission loss, and performance storage simultaneously, an efficient reliability evaluation method for multi-state phased mission systems with common bus performance sharing is proposed. The transmission loss during the processes thatperformance sharing between subsystems and transferring between phases is considered in the system model. A modified Markov model combined with the mass function is adopted to measure the precise belief degree of component states at any given moment. Then, the belief universal generating function (UGF) method based on the Dempster—Shafer evidence theory is utilized to evaluate the uncertainty of the system instantaneous availability. Finally, two case studies are carried out to demonstrate the effectiveness of the proposed method and explore the dynamic system availability under epistemic uncertainty.

A multi‐state k‐out‐of‐n:F balanced system with a rebalancing mechanism

Article

Apr 2021

Balanced systems have extensive applications in engineering fields such as new energy storage and aeronautics. The reliability analysis of such systems has been reported in literature. However, most existing studies focus on the binary‐state situation, and research on multi‐state cases and the relevant rebalancing mechanism is still limited. To fill this gap, a multi‐state k‐out‐of‐n:F balanced system with a rebalancing mechanism is studied in this paper. Both components and the system have multiple states, and all the components are required to be working in similar states to ensure that the system operates in a balanced condition. It means that the difference between the maximum and minimum component states should not exceed a threshold. Otherwise, the system is out of balance and should be rebalanced by identifying the components whose states are too high and adjusting them into lower states. A continuous‐time Markov process is used to describe the component operation process and relevant reliability indices are derived accordingly. An age maintenance strategy is also proposed and an optimization model is constructed to obtain the optimal results. Finally, numerical examples based on a product line balancing problem are presented to demonstrate the application of the proposed model.

Optimal preventive maintenance policy of the balanced system under the semi-Markov model

Article

Apr 2021
RELIAB ENG SYST SAFE

Reliability problems of new systems such as balanced systems have received more attention in recent years due to the rapid development of technology. However, the research on maintenance optimization for balanced systems is still underexplored, and the existing policies for orthodox systems cannot solve the maintenance problem of balanced systems. Thus, an optimal preventive maintenance optimization model is formulated under the semi-Markov model for a balanced system with n identical units, where each unit is subject to degradation failure and sojourn times in different function zones follow Erlang distribution with different parameters, respectively. Different from orthodox systems, the failure criterion of the balanced system is given according to the system balance degree. Namely, when any two symmetric components do not perform the same function, the system is out of balance. To avoid system unbalance, a preventive maintenance action is performed once the state difference on any two symmetric units exceeds the maintenance threshold. Finally, some numerical examples are used to demonstrate the priority of the proposed preventive maintenance policy, and the obtained results provide a good insight for repairmen to maintain components on the balanced system.

Reliability assessment for balanced systems with restricted rebalanced mechanisms

Article

Nov 2020
COMPUT IND ENG

Research on reliability modeling and assessment for balanced systems has been getting increased attention in the field of reliability and engineering due to its developments both in theory and applications. In existing research on balanced systems, forcing down and resuming units are two main ways to rebalance the system after unit failures and they are assumed to be perfectly accomplished. In this paper, two restricted rebalanced mechanisms are proposed by considering random failures of unit forced-down function and resumed function respectively with the intent of modelling actual engineering situations. Based on the two restricted rebalanced mechanisms, corresponding new balanced systems are constructed respectively. The operation mechanisms and system failure criteria of the new models are different from those of existing models. A Markov process imbedding approach is employed to analyze the system reliability and derive related probabilistic indices of the proposed models. Case studies have served the purpose to present the applicability of the new balanced systems with restricted reba-lanced mechanisms and the effectiveness of the proposed approach.

Reliability assessment of two‐level balanced systems with common bus performance sharing subjected to epistemic uncertainty

Abstract

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